Lesson 6: Solve for Unknown Angles Angles and Lines at a Point

NYS COMMON CORE MATHEMATICS CURRICULUM M1 Lesson 6 GEOMETRY

Lesson 6: Solve for Unknown Angles—Angles and Lines at a Point

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Lesson 6: Solve for Unknown Angles—Angles and Lines at a

Point

Student Outcomes

Students review formerly learned geometry facts and practice citing the geometric justifications in anticipation

of unknown angle proofs.

Lesson Notes

Lessons 1–5 serve as a foundation for the main subject of this module, which is congruence. By the end of the unknown

angles lessons (Lessons 6–8), students start to develop fluency in two areas: (1) solving for unknown angles in diagrams

and (2) justifying each step or decision in the proof-writing process of unknown angle solutions.

The “missing-angle problems” in this topic occur in many American geometry courses and play a central role in some

Asian curricula. A missing-angle problem asks students to use several geometric facts together to find angle measures in

a diagram. While the simpler problems require good, purposeful recall and application of geometric facts, some

problems are complex and may require ingenuity to solve. Historically, many geometry courses have not expected this

level of sophistication. Such courses would not have demanded that students use their knowledge constructively but

rather to merely regurgitate information. The missing-angle problems are a step up in problem solving. Why do we

include them at this juncture in this course? The main focal points of these opening lessons are to recall or refresh and

supplement existing conceptual vocabulary, to emphasize that work in geometry involves reasoned explanations, and to

provide situations and settings that support the need for reasoned explanations and that illustrate the satisfaction of

building such arguments.

Lesson 6 is Problem Set based and focuses on solving for unknown angles in diagrams of angles and lines at a point.

By the next lesson, students should be comfortable solving for unknown angles numerically or algebraically in diagrams

involving supplementary angles, complementary angles, vertical angles, and adjacent angles at a point. As always,

vocabulary is critical, and students should be able to define the relevant terms themselves. It may be useful to draw or

discuss counterexamples of a few terms and ask students to explain why they do not fit a particular definition.

As students work on problems, encourage them to show each step of their work and to list the geometric reason for

each step (e.g., “Vertical angles have equal measure”). This prepares students to write a reason for each step of their

unknown angle proofs in a few days.

A chart of common facts and discoveries from middle school that may be useful for student review or supplementary

instruction is included at the end of this lesson. The chart includes abbreviations students may have previously seen in

middle school, as well as more widely recognized ways of stating these ideas in proofs and exercises.

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Classwork

Opening Exercise (5 minutes)

Ask students to find the missing angles in these diagrams. The exercise reminds students of the basics of determining

missing angles that they learned in middle school. Discuss the facts that students recall, and use these as a starting point

for the lesson.

Opening Exercise

Determine the measure of the missing angle in each diagram.

What facts about angles did you use?

Answers may include the following: vertical angles are equal in measure; linear pairs form supplementary angles; angles

at a point sum to 𝟑𝟔𝟎°.

Discussion (4 minutes)

Discussion

Two angles ∠𝑨𝑶𝑪 and ∠𝑪𝑶𝑩, with a common side 𝑶𝑪⃗⃗⃗⃗⃗⃗ , are adjacent angles if 𝑪 belongs to the interior of ∠𝑨𝑶𝑩.

The sum of angles on a straight line is 𝟏𝟖𝟎°, and two such angles are called a linear pair. Two angles are called

supplementary if the sum of their measures is 𝟏𝟖𝟎° ; two angles are called complementary if the sum of their measures

is 𝟗𝟎° . Describing angles as supplementary or complementary refers only to the measures of their angles. The

positions of the angles or whether the pair of angles is adjacent to each other is not part of the definition.

In the figure, line segment 𝑨𝑫 is drawn.

Find 𝒎∠𝑫𝑪𝑬.

𝒎∠𝑫𝑪𝑬 = 𝟏𝟖°

The total measure of adjacent angles around a point is 𝟑𝟔𝟎° .

MP.6







51



𝑈

𝑆 𝑇

𝑉

𝑅

Find the measure of ∠𝑯𝑲𝑰.

𝒎∠𝑯𝑲𝑰 = 𝟖𝟎°

Vertical angles have equal measure. Two angles are vertical if their sides form opposite rays.

Find 𝒎∠𝑻𝑹𝑽.

𝒎∠𝑻𝑹𝑽 = 𝟓𝟐°

Example (5 minutes)

Example

Find the measure of each labeled angle. Give a reason for your solution.

Angle Angle Measure Reason

∠𝒂 𝟑𝟓° Linear pairs form supplementary angles.

∠𝒃 𝟏𝟒𝟎° Linear pairs form supplementary angles.

∠𝒄 𝟒𝟎° Vertical angles are equal in measure.

∠𝒅 𝟏𝟒𝟎° Linear pairs form supplementary angles, or vertical angles are

equal in measure.

∠𝒆 𝟗𝟕° Angles at a point sum to 𝟑𝟔𝟎°.

𝐺

𝐻

𝐾

𝐼







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Exercises (25 minutes)

Exercises

In the figures below, 𝑨𝑩̅̅ ̅̅ , 𝑪𝑫̅̅ ̅̅ , and 𝑬𝑭̅̅ ̅̅ are straight line segments. Find the measure of each marked angle, or find the

unknown numbers labeled by the variables in the diagrams. Give reasons for your calculations. Show all the steps to

your solutions.

1.

𝒎∠𝒂 = 𝟑𝟔° Linear pairs form supplementary angles.

2.

𝒎∠𝒃 = 𝟒𝟕° Consecutive adjacent angles on a line sum to

𝟏𝟖𝟎°.

3.

𝒎∠𝒄 = 𝟏𝟒° Consecutive adjacent angles on a line sum to

𝟏𝟖𝟎°

4.

𝒎∠𝒅 = 𝟒𝟗° Consecutive adjacent angles on a line sum to

𝟏𝟖𝟎°

5.

𝒎∠𝒈 = 𝟐𝟗° Angles at a point sum to 𝟑𝟔𝟎°.

MP.7







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For Exercises 6–12, find the values of 𝒙 and 𝒚. Show all work.

6.

𝒙 = 𝟖𝟎 Angles at a point sum to 𝟑𝟔𝟎°.

7.

𝒙 = 𝟑𝟎; 𝒚 = 𝟗𝟎

Vertical angles are equal in measure.

Angles at a point sum to 𝟑𝟔𝟎°.

8.

𝒙 = 𝟐𝟎 Vertical angles are equal in measure.

Consecutive adjacent angles on a line sum to

𝟏𝟖𝟎°.

9.

𝒙 = 𝟑𝟗; 𝒚 = 𝟏𝟐𝟑

Vertical angles are equal in measure. Angles

at a point sum to 𝟑𝟔𝟎°.

10.

𝒙 = 𝟖𝟎; 𝒚 = 𝟏𝟐𝟐


𝟏𝟖𝟎°.

MP.7







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11.

𝒙 = 𝟏𝟎; 𝒚 = 𝟏𝟏𝟐


𝟏𝟖𝟎°. Vertical angles are equal in measure.

12.

𝒙 = 𝟐𝟕; 𝒚 = 𝟒𝟕


𝟏𝟖𝟎°. Vertical angles are equal in measure.

Relevant Vocabulary

Relevant Vocabulary

STRAIGHT ANGLE: If two rays with the same vertex are distinct and collinear, then the rays form a line called a straight

angle.

VERTICAL ANGLES: Two angles are vertical angles (or vertically opposite angles) if their sides form two pairs of opposite

rays.

Closing (1 minute)

What is the distinction between complementary and supplementary angles?

Two angles are called complementary if their angle sum is 90°, and two angles are called

supplementary if their angle sum is 180°. The angles do not have to be adjacent to be complementary

or supplementary.

If two angles are adjacent and on a straight line, what are they called?

A linear pair

What is the angle sum of adjacent angles around a point?

The angle sum of adjacent angles around a point is 360°.

Exit Ticket (5 minutes)

MP.7







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Name Date


Exit Ticket

Use the following diagram to answer the questions below:

1.

a. Name an angle supplementary to ∠𝐻𝑍𝐽, and provide the reason for your calculation.

b. Name an angle complementary to ∠𝐻𝑍𝐽, and provide the reason for your calculation.

2. If 𝑚∠𝐻𝑍𝐽 = 38°, what is the measure of each of the following angles? Provide reasons for your calculations.

a. 𝑚∠𝐹𝑍𝐺

b. 𝑚∠𝐻𝑍𝐺

c. 𝑚∠𝐴𝑍𝐽







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Exit Ticket Sample Solutions

Use the following diagram to answer the questions below:

1.

a. Name an angle supplementary to ∠𝑯𝒁𝑱, and provide the reason for your calculation.

∠𝑱𝒁𝑭 or ∠𝑯𝒁𝑮

Linear pairs form supplementary angles.

b. Name an angle complementary to ∠𝑯𝒁𝑱, and provide the reason for your calculation.

∠𝑱𝒁𝑨

The angles sum to 𝟗𝟎°.

2. If 𝒎∠𝑯𝒁𝑱 = 𝟑𝟖°, what is the measure of each of the following angles? Provide reasons for your calculations.

a. 𝒎∠𝑭𝒁𝑮

𝟑𝟖°

b. 𝒎∠𝑯𝒁𝑮

𝟏𝟒𝟐°

c. 𝒎∠𝑨𝒁𝑱

𝟓𝟐°







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Problem Set Sample Solutions

In the figures below, 𝑨𝑩̅̅ ̅̅ and 𝑪𝑫̅̅ ̅̅ are straight line segments. Find the value of 𝒙 and/or 𝒚 in each diagram below.

Show all the steps to your solutions, and give reasons for your calculations.

1.

𝒙 = 𝟏𝟑𝟑

𝒚 = 𝟒𝟑

Angle addition postulate Linear pairs form

supplementary angles.

2. 𝒙 = 𝟕𝟖

Consecutive adjacent angles on a line sum to 𝟏𝟖𝟎°.

3. 𝒙 = 𝟐𝟎

𝒚 = 𝟏𝟏𝟎

Consecutive adjacent angles on a line sum to 𝟏𝟖𝟎°.








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Key Facts and Discoveries from Earlier Grades

Facts (With Abbreviations Used in

Grades 4–9) Diagram/Example

How to State as a Reason in an

Exercise or a Proof


(vert. s)

𝑎° = 𝑏°

“Vertical angles are equal in

measure.”

If 𝐶 is a point in the interior of

∠𝐴𝑂𝐵, then 𝑚∠𝐴𝑂𝐶 + 𝑚∠𝐶𝑂𝐵 =

𝑚∠𝐴𝑂𝐵.

(s add)

𝑚∠𝐴𝑂𝐵 = 𝑚∠𝐴𝑂𝐶 + 𝑚∠𝐶𝑂𝐵

“Angle addition postulate”

Two angles that form a linear pair are

supplementary.

(s on a line)

𝑎° + 𝑏° = 180

“Linear pairs form supplementary

angles.”

Given a sequence of 𝑛 consecutive

adjacent angles whose interiors are

all disjoint such that the angle

formed by the first 𝑛 − 1 angles and

the last angle are a linear pair, then

the sum of all of the angle measures

is 180°.

(∠𝑠 on a line)

𝑎° + 𝑏° + 𝑐° + 𝑑° = 180

“Consecutive adjacent angles on a

line sum to 180°.”

The sum of the measures of all

angles formed by three or more rays

with the same vertex and whose

interiors do not overlap is 360°.

(s at a point)

𝑚∠𝐴𝐵𝐶 + 𝑚∠𝐶𝐵𝐷 + 𝑚∠𝐷𝐵𝐴 = 360°

“Angles at a point sum to 360°.”

ao bo







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Exercise or a Proof

The sum of the 3 angle measures of

any triangle is 180°.

( sum of △)

𝑚∠𝐴 + 𝑚∠𝐵 + 𝑚∠𝐶 = 180°

“The sum of the angle measures in

a triangle is 180°.”

When one angle of a triangle is a

right angle, the sum of the measures

of the other two angles is 90°.

( sum of rt. △)

𝑚∠𝐴 = 90°; 𝑚∠𝐵 + 𝑚∠𝐶 = 90°

“Acute angles in a right triangle

sum to 90°.”

The sum of each exterior angle of a

triangle is the sum of the measures

of the opposite interior angles, or the

remote interior angles.

(ext. of △)

𝑚∠𝐵𝐴𝐶 + 𝑚∠𝐴𝐵𝐶 = 𝑚∠𝐵𝐶𝐷

“The exterior angle of a triangle

equals the sum of the two opposite

interior angles.”

Base angles of an isosceles triangle

are equal in measure.

(base s of isos. △)

“Base angles of an isosceles

triangle are equal in measure.”

All angles in an equilateral triangle

have equal measure.

(equilat. △)

“All angles in an equilateral triangle

have equal measure."







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Exercise or a Proof

If two parallel lines are intersected

by a transversal, then corresponding

angles are equal in measure.

(corr. s, 𝐴𝐵̅̅ ̅̅ || 𝐶𝐷̅̅ ̅̅ )

“If parallel lines are cut by a

transversal, then corresponding

angles are equal in measure.”

If two lines are intersected by a

transversal such that a pair of

corresponding angles are equal in

measure, then the lines are parallel.

(corr. s converse)

“If two lines are cut by a


corresponding angles are equal in

measure, then the lines are

parallel.”

If two parallel lines are intersected

by a transversal, then interior angles

on the same side of the transversal

are supplementary.

(int. s, 𝐴𝐵̅̅ ̅̅ || 𝐶𝐷̅̅ ̅̅ )


transversal, then interior angles on

the same side are supplementary.”



interior angles on the same side of

the transversal are supplementary,

then the lines are parallel.

(int. s converse)



interior angles on the same side

are supplementary, then the lines

are parallel.”

If two parallel lines are intersected by

a transversal, then alternate interior

angles are equal in measure.

(alt. s, 𝐴𝐵̅̅ ̅̅ || 𝐶𝐷̅̅ ̅̅ )


transversal, then alternate interior

angles are equal in measure.”



alternate interior angles are equal in

measure, then the lines are parallel.

(alt. s converse)



alternate interior angles are equal

in measure, then the lines are

parallel.”





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Lesson 6: Solve for Unknown Angles Angles and Lines at a Point